# Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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### Marginal density equal to zero everywhere.

I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
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### Prove Independence of a Bivariate Normal Distribution

Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent. I'm feeling pretty stumped by this question, ...
1 vote
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### Solution to bivariate normally distributed problem

Is anyone able to solve the following integral with a normal bivariate PDF, or at least to clarify if there exist a closed form solution? Thanks in advance. \frac{1}{2\pi} \int_{-\...
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### Finding the expected value of a bivariate gaussian distribution

Suppose that $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. How do I find the expected value for $\mathbb{E}[max(3.9A+B,A+3.9B)]$? I know that A and B follow the standard normal ...
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### Proof that $\mathbb{E}\bigl[1-\Phi(x_1 - \theta_1) \mid \theta_2 \leq x_2-σx_1+σθ_1\bigr]$ is increasing in $\sigma$

Numerically it appears that the following function is increasing (or at least non-decreasing) in $σ$, $$f(x_1,x_2;σ)=∫_{-∞}^{∞}∫_{-∞}^{x_2-σx_1+σθ_1}[1-Φ(x_1-θ_1)]φ(θ_1)φ(θ_2)dθ_2dθ_1$$ where $\phi$ ...
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### Finding $P(X \leq a | Y = k)$ where $X\sim\operatorname{Exp}(\lambda)$ and $Y = [X]$ where $[\cdot]$ rounds up to the nearest integer.

$Y = [X]$ where $[\cdot]$ rounds up to the nearest integer. It's given that $X \sim \operatorname{Exp}(\lambda)$. The first part asks me to show that $Y \sim \operatorname{Geo}(1-e^{-\lambda})$. The ...
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1 vote
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### Guessing a function that maximises the variance?

I'm doing a question that discusses a six-sided fair die being rolled and then discussing the numbers that appear on the top and on the side facing you. The rest of the question discusses the ...
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### In search of a bivariate distribution with nicely behaved integrals over a surface

I am looking for a joint distribution with infinite support on $R^2$, where $P(Y>max(c,aX+b))$, where $a$,$b$ and $c$ are real numbers, has a closed-form solution (without integrals). I tried many ...
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### Gaussian Random Process

I have a random process, $$Y_i=Bf_i+W_i,\quad i=0,\ldots,N$$ The RVs $B,W_1,\ldots,W_n$ are i.i.d., with $B\thicksim\mathcal{N}(0,\sigma_b^2)$ and $W_i\thicksim\mathcal{N}(0,\sigma_w^2)$. The ...
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